Periodic Reporting for period 3 - HIEXP (High Dimensional Expanders, Ramanujan Complexes and Codes)
Reporting period: 2019-08-01 to 2021-01-31
The goal of the current project is to develop high dimensional theory of expanders. We already have some applications, but the history of expander graphs teaches us that we should expect also more unexpected applications. indeed, already in the first two years of the project we came to a totally unexpected application to non-approximated groups: this is solving a first case of a series of problems which were completely untouchable till recently. We also try to get locally testable codes. If we will succeed this will be a real contribution to computer science: finding such codes is a major ( if not THE major) problem of modern error correcting codes. It represents a new paradigm: up to now all codes obtained were explicit constructions of what is known to exist by random consideration. Local testabilty is different: random codes are not locally testable!
Th emain thing we hope to do is finding locally testable codes.